Question

Compute the number of non-negative integers solutions of the following equation:

x₁ + 2x₂ + 4x₃ + 12x₄ + x₅ = n,

where 0 $\leq$ x₁ $\leq$ 3, and 1 $\leq$ x₂ $\leq$ 6.

Please show all detailed steps and reasoning.

Answer 1

Given data x_1 + 2 x_2 + 4 x_3 + 12 x_4 + x_5 = n 0 lessthanorequalto x_1 lessthanorequalto 3: 1 + x + x^2 + x^3 = 1 - x^4/1 - x 1 lessthanorequalto x_2 lessthanorequalto 6: 2 x_2 = x^2 + x^4 + ... + x^12 = x^2 (1 + (x^2)^2 + ... + (x^2)^5) = x^2 (1 - (x^2)^6)/1 - x^2 = x^2 (1 - x^12)/(1 - x) (1 + x) 4 x_3: lt x^4 + (x^4)^2 + ... = 1/1 - x^4 12 x_4: 1 + x^12 + (x^12)^2 + ... = 1/1 - x^12 x_5: 1 + x + x^2 + x^3 ... = 1/1 - x So the generating function(1 - x^4/1 - x) (x^2 (1 - x^12)/(1 - x) (1 + x)) (1/1 - x^4) (1/1 - x^12) (1/1 - x) = x^2/ (1 - x)^3 (1 + x) = x^2/(1 - x)^2 (1/(1 - x) (1 + x)) = x^2/2 (1 - x)^2 (1/1 - x +